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Power series for the hyperbolic secant
The hyperbolic secant can be expressed by the power series
Explanation
The first derivative of the hyperbolic secant is
In the outcome are only the secant and tangent. If we determine further derivatives this will remain so because (tanh x)' = sech2x, so
The point x = 0 gives sech (0) = 1 and tanh (0) = 0, so
We substitute this in the Taylor series and find
so
General format
You can write the power series as a sum
where En is the n-th Euler number.